Control Valve Tubing

A Google search for "instrument air tube sizing" will provide you with calculations (and even some web-based tube sizing calculators) to use.

The issue here is speed of response: using tubing that is too small will restrict air flow and result in a control valve that moves too slowly. In order to calculate air flow, you will need to calculate the displacement of the actuator (piston, diaphragm) and the desired stroke time to figure out how many cubic feet of air per minute is needed to move the valve. Bear in mind that you will need to compensate for the compressed air's pressure to get "standard" flow rate (e.g. SCFM). "Actual" volumetric flow rate is not the same as "Normal" or "Standard" flow rate for compressible gases.

Something else to consider aside from tubing restriction is the flow capacity of whatever pneumatic device is throttling this air to the valve. If, for example, you are using an I/P converter to directly drive air to and from the actuator, it could be that the device itself is far more restrictive than the tubing you're focused on. In other words, you could properly size all the tubing and still end up with a valve that is too slow if the supplying instrument is limited in its capacity. The manufacturer will have this data available to you either in a datasheet or via a phone call.

If there is a lot of distance between the instrument air header and the control valve, another option for reducing actuation time is to install a "capacity tank" near the valve to provide supply pressure stabilization during times of high air flow demand. If the valve's motions are infrequent (i.e. you need it to move fast, but only once in a while), a capacity tank might provide you with what you need without installing a lot of expensive large-diameter tubing.

If the speed of your valve is critical (e.g. compressor anti-surge bypass valve), you might want to take the time to run some empirical tests: set up a valve on the bench in the instrument shop and simulate different tubing scenarios using coiled lengths of plastic tubing to replicate long tube runs. Measure the valve's actual stroking speed under different conditions and see which system changes result in the greatest performance gains.